MathDB

3

Part of 2007 Nicolae Coculescu

Problems(6)

Angle bisector parallelism condition

Source:

12/13/2019
Let M,N M,N be points on the segments AB,AC, AB,AC, respectively, of the triangle ABC. ABC. Also, let P,Q, P,Q, be the midpoints of the segments MN,BC, MN,BC, respectively. Knowing that PQ PQ is parallel to the bisector of ∠BAC, \angle BAC , show that BM=CN. BM=CN.
Gheorghe Duță
Gemetryangle bisectorgeometry
Another cyclic logarithmic inequality

Source:

12/13/2019
Show that for any three numbers a,b,c∈(1,∞), a,b,c\in (1,\infty ) , the following inequality is true: log⁡abc+log⁡bca+log⁡cab≥loga2bcbc+logb2caca+logc2abab \log_{ab} c +\log_{bc} a +\log_{ca} b\ge log_{a^2bc} bc +log_{b^2ca} ca +log_{c^2ab} ab
Costel Anghel
inequalities
A certain magma of natural numbers

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12/13/2019
Determine all sets of natural numbers A A that have at least two elements, and satisfying the following proposition: \forall x,y\in A  x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A.
Marius Perianu
algebraabstract algebraDivisibilitynumber theory
Euler-like limit

Source:

12/13/2019
Let be the sequence (an)n≥0 \left( a_n \right)_{n\ge 0} of positive real numbers defined by a_n=1+\frac{a_{n-1}}{n} , \forall n\ge 1. Calculate lim⁡n→∞ann. \lim_{n\to\infty } a_n ^n .
Florian Dumitrel
real analysislimitsseuqence
Monotony conditions

Source:

12/13/2019
Consider a function f:R⟶R. f:\mathbb{R}\longrightarrow\mathbb{R} . Show that:
a) f f is nondecreasing, if f+g f+g is nondecreasing for any increasing function g:R⟶R. g:\mathbb{R}\longrightarrow\mathbb{R} .
b) f f is nondecreasing, if f⋅g f\cdot g is nondecreasing for any increasing function g:R⟶R. g:\mathbb{R}\longrightarrow\mathbb{R} .
Cristian Mangra
functionmonotonyalgebra
Another sequence defined by a primitive recursion

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12/13/2019
Let F:R⟶R F:\mathbb{R}\longrightarrow\mathbb{R} be a primitive with F(0)=0 F(0)=0 of the function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} defined as f(x)=sin⁡(x2), f(x)=\sin (x^2) , and let be a sequence (an)n≥0 \left( a_n \right)_{n\ge 0} with a0∈(0,1) a_0\in (0,1) and defined as an=an−1−F(an−1). a_{n}=a_{n-1}-F\left( a_{n-1} \right) .
Calculate lim⁡n→∞ann. \lim_{n\to\infty } a_n\sqrt{n} .
Florian Dumitrel
functionreal analysisprimitives